Question

A molecule has a rotational inertia of \(14,000 \mathrm{u} \cdot \mathrm{pm}^{2}\) and is spinning at an angular speed of \(4.30 \times 10^{12} \mathrm{rad} / \mathrm{s}\). (a) Express the rotational inertia in \(\mathrm{kg} \cdot \mathrm{m}^{2} .(b)\) Calculate the rotational kinetic energy in \(\mathrm{eV}\).

Short answer

The rotational inertia of the molecule in \(kg \cdot m^2\) is calculated in step 1. Using this value and the provided angular speed, the kinetic energy of the molecule (in Joules) is determined in step 2. The kinetic energy is then converted to electron volts in step 3. This series of conversions and calculations lead to the final answer.

Step-by-step solution

Convert unit of rotational inertia

To convert the unit of rotational inertia from \(14,000 u \cdot pm^2\) to \(kg \cdot m^2\), use these conversion factors: \(1 u = 1.660539040 \times 10^{-27} kg\) and \(1 pm = 10^{-12} m\). Therefore, by multiplying by these conversion factors, get the rotational inertia in the required unit.

Calculate rotational kinetic energy

The rotational kinetic energy (\(Ek\)) can be calculated using the formula \(Ek = \frac{1}{2} I \omega^2\), where \(I\) is the rotational inertia and \(\omega\) is the angular speed. Substituting the given values \(\omega = 4.30 \times 10^{12} rad/s\) and the converted \(I\) from step 1 into the formula, find \(Ek\) in Joules, which is the SI unit of energy.

Convert energy to electron volts

Given \(1 eV = 1.602 \times 10^{-19} J\), convert the obtained rotational kinetic energy to electron volts by dividing the energy in Joules by the conversion factor.

Key concepts

Rotational Kinetic Energy

When we talk about the rotational kinetic energy of a molecule, it's a fascinating concept that applies to objects in rotational motion. Similar to how linear kinetic energy depends on mass and velocity, rotational kinetic energy depends on rotational inertia and angular speed. The formula to find it is given by:

• Rotational Kinetic Energy: \[Ek = \frac{1}{2} I \omega^2\]

Here, \(I\) represents the rotational inertia and \(\omega\) is the angular speed.
The rotational kinetic energy provides insight into how much energy a spinning object, like a molecule, possesses due to its rotation.
This energy can be substantial, especially when considering the extremely high angular speeds at which molecules might rotate. Understanding rotational kinetic energy helps students grasp how energy conservation and mechanical systems function.
It also connects rotational dynamics with topics in physics such as energy transformations and system interactions.

Unit Conversion

Unit conversion is a critical skill in physics, allowing us to express measurements in different units to facilitate calculations and comparisons. In the original exercise, we are required to convert the unit of rotational inertia from atomic units and picometers squared (\[u \cdot pm^2\]) to the standard SI unit \(kg \cdot m^2\).
This process employs the following conversion factors:

• 1 atomic mass unit (\[1 \text{ u}\]) = \(1.660539040 \times 10^{-27}\) kg
• 1 picometer (\[1 \text{ pm}\]) = \(10^{-12}\) meters

Thus, to convert rotational inertia:

• Multiply the given value by the factors mentioned above.

This will allow us to transform the units into something more universally applicable, helping physicists and engineers solve problems effectively using a consistent unit base. Unit conversions are a fundamental tool in any scientific discipline, ensuring that data is easily interpretable and functions correctly within mathematical formulas.

Angular Speed

Angular speed signifies the rate at which an object rotates around a point or axis. In molecular physics, angular speed is a pivotal term, as it helps quantify how fast a molecule spins. It's represented by the symbol \(\omega\) and typically measured in radians per second (\[rad/s\]).
In the exercise, the given angular speed of the molecule is an impressive \(4.30 \times 10^{12} \) rad/s. Such rapid rotations are common on the molecular level, involving extremely small scales and fast dynamics.
Understanding angular speed is crucial because it directly affects the rotational energy of the object. As the angular speed increases, so does the rotational energy. It plays a role not only in molecular dynamics but also in broader applications like robotics, astrodynamics, and other fields where rotational movements are crucial.
Angular speed allows us to calculate how often an object makes a full rotation and understand the impact of this speed on the object's kinetic energy and systems it interacts with.

Molecular Physics

Molecular physics examines how molecules rotate, vibrate, and move, providing insights into their physical properties and behaviors. The discipline bridges chemistry and physics by studying molecules using principles of quantum mechanics and classical physics.
Rotational inertia and angular speed, as discussed in the exercise, are fundamental to understanding rotational dynamics at the molecular level. When molecules rotate, each has a characteristic rotational inertia that influences how it will respond to applied forces.
At such small scales, molecular rotations can significantly impact chemical reactions, spectroscopy measurements, and thermodynamics. By comprehending the way molecules handle energy through rotation, scientists can predict molecule behavior and design materials with desired properties.
Molecular physics involves detailed calculations using rigorous quantum theory modifications to account for the unique rotations and energies that arise in molecular systems. It aids in advancing our overall comprehension of molecular encounters and transforms theoretical concepts into realized scientific applications.